4The Langlands correspondence
IV Topics in Number Theory
4.4 Local Langlands correspondence
Theorem (Harris–Taylor, Henniart). There is a bijection
irreducible, admissible
representations of GL
n
(F )
←→
semi-simple n-dimensional
representations of L
F
.
Moreover,
– For n = 1, this is the same as local class field theory.
– Under local class field theory, this corresponds between ω
π
and det σ.
–
The supercuspidals correspond to the irreducible representations of
W
F
itself.
–
If a supercuspidal
π
0
corresponds to the representation
σ
0
of
W
F
, then the
essentially square integrable representation
π
=
Q
(
π
0
(
−
r−1
2
)
, . . . , π
0
(
r−1
2
))
corresponds to σ = σ
0
⊗ Sym
r−1
C
2
.
–
If
π
i
correspond to
σ
i
, where
σ
i
are irreducible and unitary, then the
tempered representation
Ind
G
P
(
π
1
⊗ ··· ⊗ π
r
) corresponds to
σ
1
⊕ ··· ⊕ σ
r
.
– For general representations, if π is the Langlands quotient of
Ind(π
1
(t
1
), . . . , π
r
(t
r
))
with each
π
i
tempered, and
π
i
corresponds to unitary representations
σ
i
of L
F
, then π corresponds to
L
σ
i
⊗ |Art
−1
F
|
t
i
F
.
The hard part of the theorem is the correspondence between the supercuspidal
representations and irreducible representations of
W
F
. This correspondence is
characterized by ε-factors of pairs.
Recall that for an irreducible representation of
W
F
, we had an
ε
factor
ε
(
σ, µ
F
, ψ
). If we have two representations, then we can just take the tensor
product
ε
(
σ
1
⊗σ
2
, µ
F
, ψ
). It turns out for supercuspidals, we can also introduce
ε
-factors
ε
(
π, µ
F
, ψ
). There are also
ε
factors for pairs,
ε
(
π
1
, π
2
, µ
F
, ψ
). Then
the correspondence is such that if π
i
correspond to σ
i
, then
ε(σ
1
⊗ σ
2
, µ
F
, ψ) = ε(π
1
, π
2
, µ
F
, ψ).
When
n
= 1, we get local class field theory. Recall that we actually have a
homomorphic correspondence between characters of
F
×
and characters of
W
F
,
and the correspondence is uniquely determined by
(i)
The behaviour on unramified characters, which is saying that the Artin
map sends uniformizers to geometric Frobenii
(ii)
The base change property: the restriction map
W
ab
F
0
→ W
ab
F
correspond to
the norm map of fields
If we want to extend this to the local Langlands correspondence, the correspond-
ing picture will include
(i)
Multiplication: taking multiplications of
GL
n
and
GL
m
to representations
of GL
mn
(ii)
Base change: sending representations of
GL
n
(
F
) to representations of
GL
n
(F
0
) for a finite extension F
0
/F
These thing exist by virtue of local Langlands correspondence (much earlier,
base change for cyclic extensions was constructed by Arthur–Clozel).
Proposition.
Let
σ
:
W
F
→ GL
n
(
C
) be an irreducible representation. Then
the following are equivalent:
(i) For some g ∈ W
F
\ I
F
, σ(g) has an eigenvalue of absolute value 1.
(ii) im σ is relatively compact, i.e. has compact closure, i.e. is bounded.
(iii) σ is unitary.
Proof. The only non-trivial part is (i) ⇒ (ii). We know
im σ = hσ(Φ), σ(I
F
) = Hi,
where Φ is some lift of the Frobenius and
H
is a finite group. Moreover,
I
F
is
normal in
W
F
. So for some
n ≥
1,
σ
(Φ
n
) commutes with
H
. Thus, replacing
g
and Φ
n
with suitable non-zero powers, we can assume
σ
(
g
) =
σ
(Φ
n
)
h
for some
h ∈ H
. Since
H
is finite, and
σ
(Φ
n
) commutes with
h
, we may in fact assume
σ(g) = σ(Φ)
n
. So we know σ(Φ) has eigenvalue with absolute value 1.
Let
V
1
⊆ V
=
C
n
be a sum of eigenspaces for
σ
(Φ)
n
with all eigenvalues
having absolute value 1. Since
σ
(Φ
n
) is central, we know
V
1
is invariant, and
hence
V
is irreducible. So
V
1
=
V
. So all eigenvalues of
σ
(Φ) have eigenvalue 1.
Since
V
is irreducible, we know it is
F
-semisimple. So
σ
(Φ) is semisimple. So
hσ(Φ)i is bounded. So im σ is bounded.